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Objective Objective To solve linear systems using To solve linear systems using elimination (adding, elimination (adding, subtracting, and multiplying). subtracting, and multiplying). Today we are focusing on Today we are focusing on solving systems by elimination solving systems by elimination when you have to multiply first when you have to multiply first Steps are on PAGE 305 Steps are on PAGE 305

Solve linear systems by Multiplication

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Page 1: Solve linear systems by Multiplication

ObjectiveObjective

To solve linear systems using elimination To solve linear systems using elimination (adding, subtracting, and multiplying).(adding, subtracting, and multiplying).

Today we are focusing on solving systems Today we are focusing on solving systems by elimination when you have to multiply by elimination when you have to multiply firstfirst

Steps are on PAGE 305Steps are on PAGE 305

Page 2: Solve linear systems by Multiplication

Equation 1 Equation 1

-2x + 5y = 13-2x + 5y = 13 Equation 2Equation 2

2x + 3y = 11 2x + 3y = 11

BELLWORKBELLWORK

Equation 1 Equation 1 2x + 3y = 11 2x + 3y = 11 Substitute value forSubstitute value fory into either of the y into either of the original equations original equations 2x + 3(3) = 112x + 3(3) = 11

2x + 9 = 112x + 9 = 11

The solution is the point (1,3). The solution is the point (1,3). Substitute (1,3) into both Substitute (1,3) into both

equations to check.equations to check.

-2(1) + 5(3) = 13-2(1) + 5(3) = 1313 = 1313 = 13

2(1) + 3(3) = 112(1) + 3(3) = 1111 = 11 11 = 11

ADDITIONADDITION

+ 8y = 248y = 24 y = 3y = 3

x = 1x = 1

EliminatedEliminated

Solution: (1,3)

Page 3: Solve linear systems by Multiplication

Could we solve this system by addition or subtraction?

6x + 5y = 19 6x + 5y = 19 2x + 3y = 52x + 3y = 5

NO!We need to do somethingBefore they cancel

Page 4: Solve linear systems by Multiplication

Section 9.4 “Solve Linear Systems Section 9.4 “Solve Linear Systems by Multiplying” p.303by Multiplying” p.303

ELIMINATION-ELIMINATION-adding or subtracting equations to obtain a adding or subtracting equations to obtain a

new equation in one variable. new equation in one variable.

Solving Linear Systems Using EliminationSolving Linear Systems Using Elimination

(1) (1) Multiply Multiply the whole equation by a constant in order to the whole equation by a constant in order to be able to eliminate a variable. be able to eliminate a variable.

(2) (2) Add or SubtractAdd or Subtract the equations to eliminate one the equations to eliminate one variable.variable.

(3) (3) SolveSolve the resulting equation for the other variable.the resulting equation for the other variable.

(4) (4) Substitute Substitute in either original equation to find the value in either original equation to find the value of the eliminated variable. of the eliminated variable.

Page 5: Solve linear systems by Multiplication

Equation 1 Equation 1

2x + 3y = 52x + 3y = 5 Equation 2Equation 2

6x + 5y = 19 6x + 5y = 19

““Solve Linear Systems by Elimination Solve Linear Systems by Elimination Multiplying First!!”Multiplying First!!”

Equation 2 Equation 2 2x + 3y = 52x + 3y = 5 Substitute value forSubstitute value fory into either of the y into either of the original equations original equations

2x + 3(-1) = 52x + 3(-1) = 5 2x - 3 = 52x - 3 = 5

The solution is the point (4,-1). The solution is the point (4,-1). Substitute (4,-1) into both Substitute (4,-1) into both

equations to check.equations to check.

6(4) + 5(-1) = 196(4) + 5(-1) = 1919 = 1919 = 19

2(4) + 3(-1) = 52(4) + 3(-1) = 55 = 55 = 5

Multiply Multiply

FirstFirst

++ -4y = 4-4y = 4

y = -1y = -1

x = 4x = 4

EliminatedEliminated

x (-3)x (-3) -6x – 9y = -15-6x – 9y = -15 6x + 5y = 19 6x + 5y = 19

Solution: (4,-1)

Page 6: Solve linear systems by Multiplication

Equation 1 Equation 1

3x + 10y = -33x + 10y = -3 Equation 2Equation 2

2x + 5y = 3 2x + 5y = 3

““Solve Linear Systems by Elimination Solve Linear Systems by Elimination Multiplying First!!”Multiplying First!!”

Equation 1 Equation 1 2x + 5y = 32x + 5y = 3 Substitute value forSubstitute value forx into either of the x into either of the original equations original equations 2(9) + 5y = 32(9) + 5y = 3

18 + 5y = 318 + 5y = 3

The solution is the point (9,-3). The solution is the point (9,-3). Substitute (9,-3) into both Substitute (9,-3) into both

equations to check.equations to check.

2(9) + 5(-3) = 32(9) + 5(-3) = 33 = 33 = 3

3(9) + 10(-3) = -33(9) + 10(-3) = -3-3 = -3 -3 = -3

Multiply Multiply

FirstFirst

++ -x = -9-x = -9

x = 9x = 9

y = -3y = -3

EliminatedEliminated

x (-2)x (-2)

3x + 10y = -33x + 10y = -3 -4x - 10y = -6 -4x - 10y = -6

Solution: (9,-3)

Page 7: Solve linear systems by Multiplication

Equation 1 Equation 1

-3x + 2y = -9-3x + 2y = -9 Equation 2Equation 2

4x + 5y = 35 4x + 5y = 35

““Solve Linear Systems by Elimination Solve Linear Systems by Elimination Multiplying First!!”Multiplying First!!”

Equation 1 Equation 1 4x + 5y = 354x + 5y = 35 Substitute value forSubstitute value forx into either of the x into either of the original equations original equations 4(5) + 5y = 354(5) + 5y = 35

20 + 5y = 3520 + 5y = 35

The solution is the point (5,3). The solution is the point (5,3). Substitute (5,3) into both Substitute (5,3) into both

equations to check.equations to check.

4(5) + 5(3) = 354(5) + 5(3) = 3535 = 3535 = 35

-3(5) + 2(3) = -9-3(5) + 2(3) = -9-9 = -9-9 = -9

Multiply Multiply

FirstFirst

++ 23x = 11523x = 115

x = 5x = 5

y = 3y = 3

EliminatedEliminated

x (2)x (2)

15x - 10y = 4515x - 10y = 45 8x + 10y = 70 8x + 10y = 70

x (-5)x (-5)

Solution: (5,3)

Page 8: Solve linear systems by Multiplication

Equation 1 Equation 1

6x + 13y = -96x + 13y = -9 Equation 2Equation 2

9x + 2y = 39 9x + 2y = 39

““Solve Linear Systems by Elimination Solve Linear Systems by Elimination Multiplying First!!”Multiplying First!!”

Equation 1 Equation 1 9x + 2y = 399x + 2y = 39 Substitute value forSubstitute value fory into either of the y into either of the original equations original equations 9x + 2(-3) = 399x + 2(-3) = 39

9x - 6 = 399x - 6 = 39

The solution is the point (5,-3). The solution is the point (5,-3). Substitute (5,-3) into both Substitute (5,-3) into both

equations to check.equations to check.

9(5) + 2(-3) = 399(5) + 2(-3) = 3939 = 3939 = 39

6(5) + 13(-3) = -96(5) + 13(-3) = -9-9 = -9-9 = -9

Multiply Multiply

FirstFirst

++ -35y = 105-35y = 105

y = -3y = -3

x = 5x = 5

EliminatedEliminated

x (2)x (2)

-18x - 39y = 27-18x - 39y = 27 18x + 4y = 78 18x + 4y = 78

x (-3)x (-3)

Solution: (5,-3)

Page 9: Solve linear systems by Multiplication

Guided PracticeGuided Practice

x + y = 2x + y = 2

2x + 7y = 92x + 7y = 9

6x – 2y = 16x – 2y = 1

-2x + 3y = -5-2x + 3y = -5

(1,1)(1,1) (-0.5, -2)(-0.5, -2)

3x - 7y = 53x - 7y = 5

9y = 5x + 59y = 5x + 5

(-10,-5)(-10,-5)

Page 10: Solve linear systems by Multiplication

HomeworkHomework

Workbook page 309-310Workbook page 309-310